The iSWAP optimality conjecture for the brickwork model

Let $n$ be the number of qubits in the brickwork model, let $t$ denote the circuit depth or number of layers, and let $\operatorname{\mathcal{T}}_{t,\mathrm{BW}}^{\operatorname{IG}}$ and $\operatorname{\mathcal{T}}_{t,\mathrm{BW}}^{\operatorname{\mathsf{iSWAP}}}$ be the corresponding moment operators for the gadget family and the iSWAP gadget. Let $\Delta(\cdot)$ denote the convergence gap. The iSWAP brickwork optimality conjecture. For $n\geq 9$,

$$\Delta\!\left(\operatorname{\mathcal{T}}_{t,\mathrm{BW}}^{\operatorname{IG}}\right)\leq \Delta\!\left(\operatorname{\mathcal{T}}_{t,\mathrm{BW}}^{\operatorname{\mathsf{iSWAP}}}\right).$$

Thus, according to the paper's numerical evidence, the iSWAP-associated gadget remains optimal for generating unitary 2-designs in the brickwork model. The claim is presented as an indication from numerical results and is left for future work.